Phase context decomposition of diagonal unitaries for higher-dimensional systems

We generalize an efficient decomposition method for diagonal operators by Welch et al. to qudit systems. The phase-context aware method focusses on cascaded entanglers whose decomposition into multi-controlled INC-gates can be optimized by the choice of a proper signed base-$d$ representation for the natural numbers. While the gate count of the best known decomposition method for general diagonal operators on qubit systems scales with $\mathcal{O}(2^n)$, the circuits synthesized by the Welch algorithm for diagonal operators with $k$ distinct phases are upper-bounded by $\mathcal{O}(n^2k)$, which is generalized to $\mathcal{O}(dn^2k)$ for the qudit case in this paper.